On the exceptional set of Lagrange’s equation with three prime and one almost–prime variables

On the exceptional ^{set of} Lagrange’s equation

with three prime ^and

^one

almost-prime ^variables

par DOYCHIN TOLEV

RÉSUMÉ. Nous considérons une version affaiblie de la conjecture

sur la représentation ^{des entiers comme somme de} quatre ^carrés de nombres premiers.

ABSTRACT. We consider an approximation ^{to the}

popular

^con- jecture ^about representations ^of integers ^{as sums of four} squares of prime ^numbers.

1. Introduction and statement of the result

The famous theorem of

Lagrange

^{states that} every

non-negative integer

n can be

represented

^as

where ~i,...,.r4 ^are

integers.

^{There is a}

conjecture,

^{which asserts that}

every

sufficiently large integer

^{n, such} that n - 4

(mod 24~,

^{can be repre-}

sented in the form

(1.1)

^with

prime

^{variables ~1, .. , x4. This}

conjecture

has ^not been

proved

^so

far,

but there are various

approximations

^{to it}

established.

We have to mention first that in 1938 Hua

[9] proved

^the

solvability

^of

the

corresponding equation

^{with five}

prime

variables. In 1976 Greaves

[3]

and later Shields

[19]

and Plaksin

[18] proved

^the

solvability

^of

(1.1)

^with

two

prime

^{and two}

integer

^variables

(in [18]

^and

[19]

^an

asymptotic

^formula

for the number of solutions was

found).

In 1994 Brfdern and

Fouvry [2]

considered

(1.1)

^with

almost-prime

^vari-

ables and

proved

that if rt is

large enough

and satisfies n - 4

(mod 24)

^then

(l.l)

has solutions in

integers

^of

type P34.

^{Here and later we denote}

by P r

any

integer

^{with no more than r}

prime factors,

^counted

according

^to

multiplicity. Recently

Heath-Brown and the author

[8] proved that,

^under

the same conditions on rt, the

equation (l.l)

^{has solutions in one}

prime

^and

three

P101 - almost-prime

variables and also in four

P25 - almost-prime

variables. These results were

sharpened slightly by

the author

~21~,

^who

lB/Ianuscrit reçu le 7janvier ^2004.

established the

solvability

^of

(1.1)

^{in one}

prime

^{and three}

P80 -

^almost-

primes and, respectively,

^{in four}

P21 - almost-primes.

There are several _papers,

published during

^{the last} years, devoted to

the

study

^{of the}

exceptional

^{set of the}

equation (1.1)

^with

^prime

^variables.

Suppose

^{that Y is a}

large

real number and denote

by El (Y)

^{the number}

of

positive integers

^{n Y}

satisfying

^{n - 4}

(mod 24)

^{and which cannot}

be

represented

^{in the form}

(1.1)

^with

^prime

^{variables ~1, ... , ~4. In 2000}

J.Liu and M.-C. Liu

[16] proved

^that

y13/15+é,

^{where e} > 0 is

arbitrarily

small. This result was

improved considerably by Wooley [22],

who established that

El(Y)

^«

y13/30+é. Recently

^{L. Liu}

[15]

established that

El (Y)

^«

^y2/5+é.

In the _paper

[22] Wooley

^{obtained otller}

interesting results, concerning

the

equation (1.1).

^{We shall state one of} them. Denote

by R(n)

^{the number}

of solutions of

(1.1)

^{in three}

prime

^{and one}

integer

variables. It is

expected

that

R(n)

^{can be}

approximated by

^the

where

C~(n)

^{is the}

corresponding singular

^series

(see [22]

^{for the}

definition).

Wooley proved

^{that the set of}

integers

n, for which

R(n)

^{fails to be close}

to the

expected value,

^is

remarkably

^{thin. More}

precisely,

^let

0(t)

^be any

monotonically increasing

^and

tending

^to

infinity

function of the

positive

variable

t,

^{such that «}

(logt)B

^{for some constant B} > 0. Let Y be a

large

^{real number and denote}

by

the number of

positive integers

Y such that

Theorem 1.2 of

[22]

^{asserts that}

We note that if the

integer n

^satisfies

Therefore,

^if

E2(Y)

^{denotes the number of}

positive integers n Y,

^satis-

fying (1.2)

^{and which cannot be}

represented

^{in the form}

(1.1)

^{with three}

prime

^{and one}

integer variables,

^then

for _any é > 0.

The _purpose of the

present

paper is to obtain an estimate of almost the

same

strength

^as

(1.3)

^{for the}

exceptional

^{set of the}

equation (1.1)

^with

three

prime

^{and one}

almost-prime

variables. We shall _prove the

following

Theorem. Let Y be a

large

real number and denote

by E(Y)

^{the number}

of positive integers

^{n Y}

satisfying

^{n - 4}

(mod 24)

^{and which cannot be}

represented

^{in the}

form

where _P1,P2,P3 are

primes

^{and x =}

P11.

^{Then we have}

As one may

expect,

^the

proof

of this result is

technically

^more

compli-

cated than the

proof

of Theorem 1.2 of

[22].

^{We use a} combination of the circle method and the sieve methods.

In the circle method

part

^we

apply

^the

approach of Wooley [22], ^adapted

for our needs. On the set of minor arcs we

apply

^{the method of Klooster-}

man, introduced in the classical _paper

[14].

^This

^technique

^was,

^actually,

applied

^also

by Wooley

in the estimation of the sums

Ti

^and

T2

in section 3 of

[22].

^{In his}

analysis, however, only Ramanujan’s

^sums appear, whiles in

our situation we have to deal with much more

complicated

sums, defined

by (8.19). Fortunately,

^{these sums differ} very

slightly

^{from sums considered}

by

Brfdern and

Fouvry [2],

^{so we can, in}

fact,

^{borrow their} result for ^our needs.

The sieve method

part

^is rather standard. We

apply

^a

weighted

^sieve

with

weights

of Richert’s type ^and

proceed

^{as in}

chapter

^{9 of Halberstam}

and Richert’s book

[4].

In _many

places

^{we omit the} calculations because

they

^{are similar to those}

in other books or papers, ^or because

they

^are standard and

straightforward.

We note that one can obtain

slightly stronger

^result

(with

^smaller power in

(1.5)

^{and with variable x}

having

^fewer

prime factors) by

^{means of more}

elaborate

computational

^work.

Acknowledgement.

^{The main} part ^{of this} paper ^was written

during

^the

visit of the author to the Institute of Mathematics of the

University

^of

Tsukuba. The author would like to thank the

Japan Society

^{of Promotion}

of Science for the financial support ^{and to} the staff of the Institute for the excellent

working

conditions. The author is

especially grateful

^{to Professor}

Hiroshi Mikawa for the

interesting

discussions and valuable comments.

The author would like to thank also Professor Trevor

Wooley

for inform-

ing

^about his paper

[22]

^and

^providing

^{with the}

manuscript.

2. Notations and some definitions

Throughout

^the paper ^{we use} standard number-theoretic notations. As

usual, p(n)

^denotes the M6bius

function,

is the Euler

function,

is the number of

prime

^{divisors of} n, counted

according

^to

multiplicity,

T(n)

^is the number of

positive

divisors of n. The

greatest

^{common divisor}

and, respectively,

^{the least common}

multiple

^{of the}

integers

^{MI, M2 are}

denoted

by (m1, m2)

^and

[m1, m2]. However,

^{if u and v are} real numbers then

(u, v)

^means the interval with

endpoints u

^{and v. The}

meaning

^is

always

clear from the context. We use bold

style

^{letters to denote four-}

dimensional vectors. The letter _p is reserved for

prime

^numbers.

If p

> 2

then -

stands for the

Legendre symbol.

^{To denote summation over the}

positive integers

^{n Z we write}

Furthermore, ~x(~,), respectively,

¿x(q)*

^{means that x} runs over a

complete, respectively,

^reduced

system

^of residues modulo _q.

By ~a~

^{we denote the}

integer ^part

^{of the} real number _a,

e(~) - e"?"

and

eq(a)

⁼

e(a/q).

We assume that E > 0 is an

arbitrarily

^small

positive

^{number and A}

is an

arbitrarily large number; they

^can take different values in different formulas. Unless it is not

specified explicitly,

^{the constants in the 0 -}

terms and « -

symbols depend

^{on E} and A. For

positive

^{U and V we write}

!7 x V as an abbreviation of U « V « U.

Let N be a

suffciently large

real number. We define

where E > 1 is a

large

constant, ^{which we shall}

specify

^later.

To

apply

^{the sieve method we need} information about the number of solutions of

(1.4)

ⁱⁿ

integers

^x

lying

ⁱⁿ arithmetical

progressions

^{and in}

primes

pl, p2, p3. For technical reasons we attach

logarithmic weights

^to

the

primes

^{and a smooth}

weight

^{to the variable x. More}

precisely,

^we

consider the function

and let

For _any

integer n

^E

(N/2, N]

^{and for} any

squarefree integer k,

^{such that}

(k, 6)

⁼

^1,

^{we define}

We expect ^{that this}

quantity

^{can be}

approximated,

^{at least on} average,

by

the

expression

which arises as a mean term when we

apply formally

the circle method.

The

quantity K(n)

^{from the}

right-hand

^{side of}

(2.6)

^{comes from the}

singular integral

^{and is defined}

by

Having

^{in mind}

(2.3)

^{we see that}

Furthermore, 6 (n, ~;)

^{comes from the}

singular

series and is defined

by

where

We shall consider

C‘~(7t, k)

^{in detail in the next section.}

3. Some

properties

^{of the sum}

6(n, k)

It is not difficult to see that the function

f (q, n,1~),

^defined

by (2.10),

^is

multiplicative

^with

respect

to q. We shall

compute

^{it for} q ⁼

p~.

From this

point

^{onwards we assume that the}

integers

^{n and k}

satisfy

Then we have

Furthermore, if p

> 2 is a

prime,

^then

and

where the

quantities hj

^{are defined}

^by

The

proof

^{of formulas}

(3.2) - (3.8)

^{is standard and uses}

only

^{the basic}

properties

^{of the Gauss sums}

(see

^Hua

[10], chapter 7,

^for

example).

^We

leave the verification to the reader.

From

(3.2) - (3.8)

^we

easily get

This estimate

implies

^{that the series}

(2.9)

^is

absolutely convergent.

^We

apply

^Euler’s

identity

^{and we use}

(3.1) - (3.5)

^{to obtain}

From this formula and

(3.4),

^{after some}

rearrangements,

^we

get

where

and where

From

(3.1), (3.5) - (3.8), (3.11)

^and

(3.13)

^we obtain the estimates

and

where the constant in the 0 -term in the last formula is absolute. We leave the _easy verification of formulas

(3.14) - (3.17)

^{to the reader.}

Let us note that from

(3.10), (3.12)

^and

(3.16)

^{it follows}

which _we, of _course,

expect, having

in mind the definition

(2.5)

^of

I(n, k)

and the conditions

(3.1).

4. Proof of the Theorem

A central role in the

proof

of the Theorem

plays

^the

following Proposi- tion,

^{which asserts} that the difference between the

quantities I(n, k)

^and

E(n, ~ ),

^defined

by (2.5)

^and

(2.6),

^{is small on average with}

^respect

^{to n}

and k.

Proposition. Suppose

^{that the set F consists}

of integers

^{n E}

(N/2, N],

satisfying

^the congruence ^{n -} 4

(mod 24),

^{and denote}

by

^{F the}

cardinality of

^{~’. Let}

q(k)

^{be a real valued}

function, defined

^{on the set}

of positive integers

and such that

and

Suppose

^{also that}

for

^{some constant}

and conszder the sum

Then we have

The constant in

Vinogradov’s symbol depends only

^{on the constants}

6, E, included, respectively,

ⁱⁿ

(2.1), (2.2)

^and

(4.3).

We shall _prove the

Proposition

in sections 5 - 8. In this section we shall

use it to establish the Theorem.

Let ,~’ be the set of

integers n

^E

(N/2, N] satisfying

^{n - 4}

(mod 24)

^and

which cannot be

represented

in the form

(1.4)

^with

primes

Pi, P2, P3 and with r ⁼

P11.

Let F be the

cardinality

^{of .~. We} shall establish that

Obviously,

^this

implies

^{the estimate}

( 1.5) .

To

study

^the

equation (1.4)

^{with an}

almost-prime

^{variable x we}

apply

^a

weighted

^{sieve of Richert’s}

type.

Let _{q, v,}

vl, 9

^{be constants such that}

Denote

and

Consider the sum

where

It is clear that

where

T1

^{is the} contribution of the terms for which > 0. We decom- pose

f1

^as

(4.13)

where

F2

^{is the} contribution of the terms with x == 0

(mod p2)

^{for some}

prime

p ^E

[z, zl)

^{and where}

F3

^{comes from the other terms.}

We note that the congruence condition ^{n -} 4

(mod 24)

^{and the size}

conditions on pi in the domain of summation in

(4.10) imply

^{that there}

are no terms with

(6, x)

> 1 counted in T

and, respectively,

ⁱⁿ

r3.

^Hence

the condition

(~, P)

^{= 1 from the domain of} summation in

(4.10)

^{can be}

replaced by (x. 6P)

^{= 1.}

Furthermore,

^{if r is}

squarefree

^with respect ^{to the}

prime

^numbers p ^E

[z, zi),

^if

(~, 6P)

^{= 1 and}

A(r)

>

0,

^then

For

explanation

^{we refer the reader to} Halberstam and Richert

[4], chap-

ter

9, p.256.

From this

point

^{onwards we assume also that}

Then

using (4.14), (4.15)

^{and the} definition of the set 17 we conclude that the sum

F3

^is

empty,

^i.e.

Indeed,

^{if this were not} true, ^{then for some n E .~ the}

equation (1.4)

^would

have a solution _P1,P2,P3, x with x

satisfying (4.14). However,

^{this is not}

possible

^{due to}

(4.15)

^and the definition of T.

It is _{easy to} estimate

F2

from above. We have

and, having

^{in mind}

(4.7)

^and

(4.8),

^we

get

From

(4.12), (4.13), (4.16)

^and

(4.17)

^it follows that

We shall now estimate T from below.

Using (4.10)

^and

(4.11)

^{we repre-}

sent it in the form

where

and where

F5

^{comes from} the second term in the

right-hand

^{side of}

(4.11).

Changing

^{the order of summation we}

get

To find a non-trivial lower bound for r we have to estimate

r4

^{from below} and

r5

from above. We shall

apply

^{Rosser’s sieve}

(see

^Iwaniec

[12], [13]).

First we

get

^rid of the condition

(x, P)

^{= 1 from the domains of summa-}

tion in

(4.20)

^and

(4.21) by introducing

^the

weight

Denote

by ~-(d)

^the lower Rosser function of order ^D and for each

prime

p ^E

[z, zl)

^denote

by A) (d)

^{the upper} Rosser function of order

D/p. They satisfy

and

Furthermore,

^let

f (s)

^and

F(s)

^{be the} functions of the linear sieve. We consider

separately

^{the cases}

5 I nand 5 f n

^{and use}

(3.15) - (3.17), (4.7) -

(4.9)

^{to find that}

where

For the definition and

properties

of Rosser’s functions and the functions

f (s), F(s)

^{as well as for}

explanation

^of

(4.23) - (4.27)

^{we refer the reader}

to Iwaniec

[12], ~13~.

Consider the sum

F4.

^From

(2.5), (4.20), (4.22), (4.23)

^and

(4.25)

^we get

where

and where

Consider now

F5 - Using (4.21), (4.22)

^and

(4.25)

^{we find that}

where

r7

^{comes from the}

quantity

^{from the}

right-hand

^{side of}

(4.25).

Changing

^{the order of summation and}

applying (2.5), (4.9)

^and

(4.24)

^we

write

IF7

^{in the form}

where

Using (4.8), (4.9), (4.23), (4.24), (4.30)

^and

(4.32)

^{we see that both func-}

tions

q’(k)

^and

q"(k) satisfy

the conditions

(4.1)

^and

(4.2).

We consider the sums

and

using

^the

Proposition

^we conclude that

According

^to

(4.19), (4.29), (4.31)

^and

(4.34)

^{we have}

Consider more

precisely

^{the sums}

F8

^and

r9.

^From

(2.6), (3.10), (3.18), (4.30)

^and

(4.33)

^it follows that

and, respectively,

^we

apply (2.6), (3.10), (3.18)

^and

(4.33)

^to

get

Furthermore, according

^to

(3.12), (4.24)

^and

(4.32)

^{we have}

From

(2.8), (3.14) - (3.17), (4.26) - (4.28)

^and

(4.36) - (4.38)

^{we obtain}

where

Now we

apply

^the

arguments

^of Halberstam and Richert

~4~, ^{chapter 9,}

p. 246 and we find

where

We

specify

^{the constants included}

by

It is _{easy to} see that

they satisfy

the conditions

(4.7)

^and

(4.15).

^Further-

more,

using

Lemma 9.1 of Halberstam and Richert

[4],

^{we can}

^verify

^that

if _{q, v, vl} and 0 are

specified by (4.41),

^then

From

(2.8), (3.14), (4.28), (4.39), (4.40)

^and

(4.42)

^we

get

where Co > 0 is a constant.

Inequalities (4.18), (4.35)

^and

(4.43) imply

We are now in a

position

^to

apply

^{the main idea of}

Wooley ~22~.

^Since

E > 1 we can omit the second term from the

right-hand

^{side of}

(4.44).

^We

get

which

implies

^{the estimate}

(4.6)

^{and proves the Theorem.}

5. The

proof

^{of the}

Proposition

^-

beginning

We represent ^{the sum}

I(n, k;),

^defined

by (2.5),

^{in the form}

where

The

integration

ⁱⁿ

(5.1)

^{can be taken over any} interval of

length

^one

and,

in

particular,

^over

, _ _ "1 - - " .

We

represent Jo

^{as an union of}

disjoint Farey

^intervals

where

and where

q’

^and

q"

^are

specified by

the conditions

(5.4)

^X

q + q’, q + q" aq’ ==

¹

(mod q) , aq" - -1 (mod q) (for

^{more details we refer the}

reader,

^for

example,

^to

Hardy

^and

Wright [5], chapter 7).

Consider the set

where

It is clear that if

1 a q P, (a, q)

⁼

^1,

^then

91(q, a)

^c

B(q, a),

^hence

we can represent

Jo

^{in the form}

where

Hence we have

where

From

(4.4), (5.10)

^and

(5.11)

^{we see that the sum E can be}

represented

as

where

To prove the

Proposition

^{we have to estimate the sums}

£1 and £2.

^{We shall}

consider

£1

in section 6 and

£2 -

in sections 7 and 8.

6. The estimation of the sum

~l

We

apply

^{Lemma 3.1}

of Wooley [22],

^{which is based on} the earlier result of

Bauer,

^{Liu and Zhan}

[1].

^{It states that if the set .M is defined}

^by (5.5),

then for _any

integer h

^{such that 1 h}

N,

^{we have}

where

Using (5.2)

^and

(5.11)

^{we write}

⁷ⁱ

^{in the form}

We now

apply (2.3), (2.4)

^and

(6.1)

^to

get

11B I t,,,

where

We note

that,

^{due to} the choice of our

weight w(x),

^{it is} not necessary to consider

analogs

of formula

(6.1)

^for

non-positive integers h,

^{as it was done}

in

[22].

Using (4.2), (5.13)

^and

(6.3)

^{we find}

where

Consider first

£i2).

^{* It was} established ⁱⁿ

[22]

^that

so,

using (2.1), (6.5), (6.7)

^and

(6.8)

^{we find}

where

It is clear that

Applying

^{Theorem 3 of Hua}

[11]

^{we find}

for some absolute constant B > 0. From

(6.9)

^and

(6.10)

^{we conclude that}

if E > B +

1,

^{which we shall assume, then}

Consider ^now the sum

£i1).

^{First we}

^{apply (6.2)}

^and

^(6.4)

^{to write the}

expression l(l)

^{in the form}

where

Furthermore,

^{we have}

where

Obviously,

^if

(k, q) t

^{m, then W* = 0. If}

(k, q) I

^{m, then there exists}

unique h

⁼

(mod [k, q]),

^{such that the}

^system

^of congruences in the domain of summation of

(6.14)

^is

equivalent

^{to r m h}

(mod [k, q]).

^We

apply

Poisson’s summation formula and use

(2.3)

^and

(2.4).

^{After some}

calculations we

get

1 then we

integrate

^{the last}

integral by

parts ^{m times.}

Having

ⁱⁿ

mind the conditions k

D, q

^{P and}

using (2.1)

^and

(4.3)

^{we find that}

the

integral

^is

where the constant in

Vinogradov’s symbol depends only

^{on m. From this}

observation we conclude that the contribution to

W*, coming

^{from the}

terms

1,

^is

negligible.

^More

precisely,

^{we have}

where

K(n)

^{is defined}

by (2.7)

^{and where the constant A} > 0 is

arbitrarily large.

From

(2.10), (2.11), (6.12), (6.13)

^and

(6.15)

^we

get

It remains to take into account also

(2.6), (2.8), (2.9)

^and

(3.9)

^{and we find}

This formula and

(2.1), (4.2), (6.7) imply

We note that _any,

arbitrarily

^small

positive

^{value of the} constant 6 from the definition of

P,

suffices for the

proof

^{of the} last estimate. This

happens

wherever P _occurs, so any progress in

obtaining asymptotic

formulas of

type (6.1)

^for

^larger

^{sets of}

major

^{arcs is not relevant to our}

problem.

Finally,

^from

(6.6), (6.11)

^and

(6.16)

^{we obtain the estimate}

7. The estimation of the ^sum

62

Using (5.11)

^and

(5.13)

^we

^represent

^{the sum}

⁹²

^{in the form}

where

Applying

^H61der’s

inequality

^we

get

where

For

T1

^we

apply

^{the well} known estimate

,,1

Consider

Tz.

^From

(7.1)

^and

(7.3)

^we

^get

where

To estimate the first term from the last line of

(7.5)

^we

^apply

^{the in-}

equality

Its

proof

^{is very similar to the}

proof

of formula

(4.3)

from author’s _pa- per

[20],

^{so we omit it.}

In the next section we shall estimate

~(l)

^{and we shall} prove that

From

(2.1), (4.3), (7.5), (7.7)

^and

(7.8)

^{we obtain}

Applying (7.2), (7.4)

^and

(7.9)

^we

get

It remains to combine

(5.12), (6.17)

^and

(7.10)

^{and we} find that the estimate

(4.5) holds,

^which proves the

Proposition.

8. The estimation of

For _any

integers

^{a, q,}

satisfying 1 a q X, (a, q)

⁼

1,

^{we consider}

the set

9R(q, a),

^defined

by

where the

integers q’

^and

q"

^are

specified by (5.4). Using (5.3), (5.6), (5.8), (5.9), (7.6)

^and

(8.1)

^{we find that}

X,

^{then we have}

where

Formula

(8.3)

^{is a}

special

^case of Lemma 12 from the _paper

[8] by

^Heath-

Brown and the author.

From

(7.1)

^and

(8.3)

^{it follows that the}

integrand

ⁱⁿ

(8.2)

^{can be repre-}

sented in the form

where

and where _n, k and u are four dimensional vectors with

components,

^re-

spectively,

ni,

ki

^and ui, i ⁼

1, 2, 3, 4.

We substitute the

expression (8.6)

^{for the}

integrand

ⁱⁿ

(8.2). Then,

ⁱⁿ

order to

apply

Kloosterman’s

method,

^we

change

the order of

integration

and summation over a. We find that

where

To

proceed

^{further we have} to express the condition

a) 3 /3

^{in more}

convenient form. We do this in a standard _way

by introducing

^{a function}

Q(v,

q,

~3),

defined for

integers

q, ^v such that 1 _q

X, -q/2

^v

q/2

and real numbers

{3

^E

[_(qX)-l, (qX)-l].

^{For fixed v and} q this function is

integrable

^with respect ^to

/3

^and

satisfy

Furthermore,

^if

(a, q)

⁼

1,

^{if a}

(mod q)

^{is defined}

by

^{aa - 1}

(mod q)

^and

if q’ and q"

^are

specified by (5.4),

^then

A construction of ^a function with these

properties

^is

available,

^for

example,

in Heath-Brown

[7],

^{section 3.}

Using (8.13)

^{we can express} the condition

a) 3 j3

^{from the domain}

of summation in

(8.10).

If P

q X, then, according

^to

(8.1), (8.11)

^and

(8.13),

^{we find that}

N

If q

^{P then we}

integrate

^over

hence, having

^{in mind}

(8.1),

^{we see that} the condition

OO1(q, a) 3 /3

ⁱⁿ

(8.10)

is

equivalent

^to

(

²⁰¹³

., ..] 1

³

^0.

^{Therefore we can use}

^{again (8-13)}

is

equivalent {3.

^{Therefore we can use}

again (8.13)

to obtain

(8.14).

We conclude that

~(l)

^{can be} written in the form

where

Using (4.1), (4.2), (8.12)

^and

(8.15)

^{we find that}

, ,- ,

where

E’

^means that the summation is taken over

squarefree odd ki

^and

Consider the sum W. From

(8.4), (8.8)

^and

(8.16)

^{we see that it can be}

written as

In this form the sum W is _very similar to a sum, considered

by

^Brfdern

and

Fouvry [2]. ^Applying

^the

method, developed

^{for the}

proof

^{of Lemma 1}

from

[2],

^we find that if

~2

^are

squarefree

^odd

integers,

^then

Consider the

quantity T,

^defined

by (8.18). Obviously

Applying (8.5), (8.7), (8.9), (8.21)

^{and Lemma}

10(ii)

^of

[8]

^we

^get

If n ⁼ 0 and P

q X,

^{then we use}

(8.5), (8.9), (8.21)

^and

apply

^the

well known estimate

We

get

If n ⁼ 0 and _q

P,

^{then we use}

(2.2), (8.11), (8.18)

^and

(8.23)

^{to obtain}

From

(8.17)

^{we find that}

where is the contribution of the terms from the

right-hand

^{side of}

(8.17)

such that

n ~ 0, ’b2

^{comes from the terms with n = 0 and P}

q ~

^X

and, finally, ~3

^{comes from the terms for which n = 0}

and q

^P.

To estimate we use

(8.20)

^and

(8.22)

^{and we}

get

After some standard

calculations,

^{which are} very similar to those in sec-

tion 5 of

[8],

^{we obtain}

We leave the verification of this estimate to the reader.

For

~2

^we

apply (8.20)

^and

(8.24)

^to

^get

To estimate

Q3

^we

apply, respectively, (8.20)

^and

(8.25)

^{and we find}

, - , ’I. , - , .

The estimate

(7.8)

^{is a} consequence of

(8.26) - (8.29).

This _proves the

Proposition

^{and now the}

proof

^{of the Theorem is com-}

plete.

Added in

proof:

^Two

interesting results, concerning

^the

quantity E1 (Y),

defined in section

1, appeared

^{since the} present paper ^was submitted for

publication. J.Liu, Wooley

^{and Yu}

[17]

established the estimate

E1 (Y)

^«

Y’18+6

and _very

recently

Harman and Kumchev

[6] proved

^that

El (Y)

^GC

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Doychin ^TOLEV

Department of Mathematics Plovdiv University ^"P. Hilendarski"

24 "Tsar Asen" str.

Plovdiv 4000, Bulgaria

E-mail : dtolev@pu.acad.bg